In this screencast, I want to do a derivation of the heat diffusion equation in cylindrical
coordinates. And the reason you might come across this equation, or the reason you might
find it useful, is that you could be faced with a problem where you need to calculate
or derive the temperature profile for a cylindrical geometry. So as an example, you might have
a copper wire with an electrical current flowing through it, and then there's some convective heat
transfer on the exterior, and the temperature of the interior of this is greater than the
temperature at the exterior of it. Another example you might come across would be the
temperature distribution across the walls of a pipe. So you might have hot water flowing
through the center of the pipe, and the exterior of the pipe is cooler than the interior, and
in both cases you can use this equation to derive those temperature profiles. While it
may not look like it, all this equation is is an energy balance around a differential
element. And what it says is that the rate at which thermal energy is being stored within
a small volume is equal to the rate at which energy is entering, minus the rate at which
energy - thermal energy - is leaving, plus any rate of thermal energy generation. So
thermal energy could be generated, for example, as an electric current or maybe a chemical
or a nuclear reaction, for example. Three axes that we need in cylindrical geometry:
The first would be the axial dimension, which we call z in this case, so z is the length
along this rod. The second is the radial direction, or the distance from the center of the rod,
and the third is this angular direction, which we call phi, which gives us the angular position
from some starting point. So on the right what I've drawn is a geometric depiction of
a differential control volume. Vertically in this case, I've got my z axis, in the radial
direction, I've got this variable r, and in the angular direction, we use a variable phi.
So you can see the three of those, we see the variable r, phi, and z in this differential
equation, which we'll derive. What I've drawn is a differential control element, or differential
control volume. it's characterized by the dimensions, say, dz in the vertical dimension,
and dr in the radial direction, and a distance rd(phi) in the angular direction. Now we've
got r times some change in angle, this r is in front for dimensional consistency. Let's
examine the net flow of energy into this control volume. We'll think about the three different
dimensions. And in the vertical direction, we can call it qz entering the control volume
from the bottom. Energy flowing in from the back, in the radial direction, we'll call
q dot r, and there's energy flowing in in the angular direction, or the direction of increasing
phi, into this face on the left, and we'll call that q dot phi. So that was energy coming
in from the three faces, now what I'm showing is energy leaving. I've got energy leaving
in the radial direction on this front face, so leaving in the direction of increasing
r, leaving in the axial direction, or the direction of increasing z, this top face right
here, and then on this back face, leaving in the angular direction which I call q dot phi
+dphi. So those are the way energies can be conducted into or out of the control volume,
and the third term we need to consider is the rate of thermal energy generation. The
rate of thermal energy generation is this term q dot multiplied by the volume of this
control element. So we've got q dot times a volume, or we could say q dot times, let's say
we want dr times dz, the vertical dimension, multiplied by r dphi. And this is analogous
to a rectangular geometry, where we say the length times the width times the height, so
in this case r dphi is the width, we've got the height of it, dz, and the length of it,
dr. And finally, the rate at which thermal energy is being stored is proportional to
the heat capacity and the rate the temperature is changing. So we take the mass, or the density
times this volume, dr times rdphi *dz. Density times a volume is a mass, multiplied by the
heat capacity, times the rate at which temperature is changing. So here I've written it all out.
I've got the energy stored term, the energy coming in, the energy coming out through the
other three faces, and the energy being generated. One thing we need to figure out is, how does
the heat leaving at q(r+dr) compare to the heat entering at q(r). So how does the heat
leaving from this face, this front face, compare to the amount of heat coming in from this
back face. We can do it for all three dimensions, the r, the phi, and the z dimension, by way
of Taylor series expansions, which I've done down here. So with these expansions, we can
plug in values now for q(r+dr), and the phi and z components of it. When I make this substitution,
what we'll find is that these q(r)'s will drop out because of the negative sign, and
the q(phi)'s just as well, and the q(z)'s will also drop out because of the negative
sign in the expression. So here's what we're left with after making the substitution. On
the left is the energy storage term, on the right, the last term is the energy generation
term, and these three terms now in the middle represent the net rate at which energy is
entering the control volume. What we need to do now is show how the flow of energy in
the r, phi, and z directions, q(r), q(phi) and q(z), how can we relate those to the temperature?
And we can do that by looking at Fourier's Law. Let's examine the flow of energy in the
radial direction first. What I've written is, q(r) is equal to -kdT/dr times r*d(phi)*d(z).
And what I'm writing is simply, the total flow of energy, or the net flow of energy,
in the r direction is equal to the flux, which is -k*dT/dr, the flux times this cross-sectional
area, or I should say the area of this face, which is r*d(phi), this length, times d(z),
or this height, but the area in the back or the area in the front. And I could do something
analogous for the angular direction, this phi direction, and in the z direction. So
in the phi direction, I'm talking about the area drdz, so this area here, and this area
in the back of it, and in the z or axial direction, I'm talking about the area on the bottom and
the top of our control volume. Note the presence of r in the denominator of the phi direction,
and the presence of that r is to make the equation dimensionally homogeneous, so we're
talking about a differential change in the distance in this angular direction, so we've
got r*d(phi). So let's make the substitution, we'll put in for q(r), q(phi), and q(z). So
here's what we're left with after making the substitution: we've got the energy stored
term, the three net energy transport into and out of the control volume now in terms
of temperature, and finally the energy generation term. And finally the last thing to do is
to divide by this dr times r d(phi) d(z), divide both sides of the equation. It'll fall out
in the energy generation term, it'll fall out in the energy storage term, and we'll
see some combination of it in the net energy coming in minus energy leaving.
And so here you have it, the heat diffusion equation in cylindrical coordinates.